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Theorem adjadd 29080
Description: The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjadd ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))

Proof of Theorem adjadd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmadjop 28875 . . 3 (𝑆 ∈ dom adj𝑆: ℋ⟶ ℋ)
2 dmadjop 28875 . . 3 (𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)
3 hoaddcl 28745 . . 3 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
41, 2, 3syl2an 493 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5 dmadjrn 28882 . . . 4 (𝑆 ∈ dom adj → (adj𝑆) ∈ dom adj)
6 dmadjop 28875 . . . 4 ((adj𝑆) ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
75, 6syl 17 . . 3 (𝑆 ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
8 dmadjrn 28882 . . . 4 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
9 dmadjop 28875 . . . 4 ((adj𝑇) ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
108, 9syl 17 . . 3 (𝑇 ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
11 hoaddcl 28745 . . 3 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
127, 10, 11syl2an 493 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
13 adj2 28921 . . . . . . . 8 ((𝑆 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
14133expb 1285 . . . . . . 7 ((𝑆 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
1514adantlr 751 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
16 adj2 28921 . . . . . . . 8 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
17163expb 1285 . . . . . . 7 ((𝑇 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1817adantll 750 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1915, 18oveq12d 6708 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
201ffvelrnda 6399 . . . . . . 7 ((𝑆 ∈ dom adj𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
2120ad2ant2r 798 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆𝑥) ∈ ℋ)
222ffvelrnda 6399 . . . . . . 7 ((𝑇 ∈ dom adj𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2322ad2ant2lr 799 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
24 simprr 811 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25 ax-his2 28068 . . . . . 6 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
2621, 23, 24, 25syl3anc 1366 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
27 simprl 809 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28 adjcl 28919 . . . . . . 7 ((𝑆 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑆)‘𝑦) ∈ ℋ)
2928ad2ant2rl 800 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑆)‘𝑦) ∈ ℋ)
30 adjcl 28919 . . . . . . 7 ((𝑇 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑇)‘𝑦) ∈ ℋ)
3130ad2ant2l 797 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑇)‘𝑦) ∈ ℋ)
32 his7 28075 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((adj𝑆)‘𝑦) ∈ ℋ ∧ ((adj𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3327, 29, 31, 32syl3anc 1366 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3419, 26, 333eqtr4rd 2696 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
357, 10anim12i 589 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ))
36 hosval 28727 . . . . . . . 8 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
37363expa 1284 . . . . . . 7 ((((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3835, 37sylan 487 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3938adantrl 752 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
4039oveq2d 6706 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)) = (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))))
411, 2anim12i 589 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42 hosval 28727 . . . . . . . 8 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
43423expa 1284 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4441, 43sylan 487 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4544adantrr 753 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4645oveq1d 6705 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
4734, 40, 463eqtr4rd 2696 . . 3 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
4847ralrimivva 3000 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
49 adjeq 28922 . 2 (((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦))) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
504, 12, 48, 49syl3anc 1366 1 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690   + caddc 9977  chil 27904   + cva 27905   ·ih csp 27907   +op chos 27923  adjcado 27940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-hilex 27984  ax-hfvadd 27985  ax-hvcom 27986  ax-hvass 27987  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991  ax-hvdistr2 27994  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his2 28068  ax-his3 28069  ax-his4 28070
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-2 11117  df-cj 13883  df-re 13884  df-im 13885  df-hvsub 27956  df-hosum 28717  df-adjh 28836
This theorem is referenced by: (None)
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